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 May 4th, 2009, 01:02 PM #1 Newbie   Joined: Mar 2009 Posts: 9 Thanks: 0 Finding the minimal polynomial if f = x^3 + 2X+1 is the minimal polynomial of alpha, then what is the minimal polynomial of alpha^2? I have no idea how to solve that. A hint pls. May 9th, 2009, 09:24 PM #2 Senior Member   Joined: Jul 2008 Posts: 144 Thanks: 0 Re: Finding the minimal polynomial It's impossible, I think. May 9th, 2009, 11:33 PM   #3
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Re: Finding the minimal polynomial

Quote:
 Originally Posted by lambda It's impossible, I think.
Hardly. I'm making (thought not a very bold one) the assumption that this f(x) is a polynomial over Z, so this implies that the roots of f(x) are algebraic integers. It is a theorem (and not a trivial one at all!) that the algebraic integers form a ring and there is a method that is based on using eigenvalues of a matrix to compute the minimal polynomial of of ab, for some algebraic integers a and b. I'm a little rusty on this method and will have to look it over and figure it out, but impossible--definitely not. May 15th, 2009, 08:56 AM #4 Newbie   Joined: Mar 2009 Posts: 9 Thanks: 0 Re: Finding the minimal polynomial Thanks for the replies but I solved the problem myself by looking at the powers of alpha^2. alpha^n can always be written as lin. combination of elements of the basis {0,alpha,alpha^2} if I do so, then I see that a^6 is a linear combination of 1,a^2,a^4 and so by solving the resulting linear equations I found the minimal polynomial. Tags finding, minimal, polynomial Thread Tools Show Printable Version Email this Page Display Modes Linear Mode Switch to Hybrid Mode Switch to Threaded Mode Similar Threads Thread Thread Starter Forum Replies Last Post Artus Linear Algebra 6 November 9th, 2012 09:19 AM metder Linear Algebra 0 November 23rd, 2009 02:17 PM TTB3 Abstract Algebra 3 April 15th, 2009 04:32 AM TimmyTimTim Abstract Algebra 0 March 18th, 2008 07:11 PM Jazz24 Abstract Algebra 1 March 6th, 2008 05:13 AM

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